Multipliers of Improper Similitudes

For a central simple algebra with an orthogonal involution (A, σ) over a field k of characteristic different from 2, we relate the multipliers of similitudes of (A, σ) with the Clifford algebra C(A, σ). We also give a complete description of the group of multipliers of similitudes when deg A ≤ 6 or when the virtual cohomological dimension of k is at most 2.


Introduction
A. Weil has shown in [22] how to obtain all the simple linear algebraic groups of adjoint type D n over an arbitrary field k of characteristic different from 2: every such group is the connected component of the identity in the group of automorphisms of a pair (A, σ) where A is a central simple k-algebra of degree 2n and σ : A → A is an involution of orthogonal type, i.e., a linear map which over a splitting field of A is the adjoint involution of a symmetric bilinear form.(See [7] for background material on involutions on central simple algebras and classical groups.)Every automorphism of (A, σ) is inner, and induced by an element g ∈ A × which satisfies σ(g)g ∈ k × .The group of similitudes of (A, σ) is defined by that condition, GO(A, σ) = {g ∈ A × | σ(g)g ∈ k × }. 1 The first author gratefully acknowledges the generous support of the Université catholique de Louvain, Belgium and the ETH-Z, Switzerland.
2 Work supported in part by the European Community's Human Potential Programme under contract HPRN-CT-2002-00287, KTAGS.The second author is supported in part by the National Fund for Scientific Research (Belgium).

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The map which carries g ∈ GO(A, σ) to σ(g)g ∈ k × is a homomorphism µ : GO(A, σ) → k × called the multiplier map.Taking the reduced norm of each side of the equation σ(g)g = µ(g), we obtain Nrd A (g) 2 = µ(g) 2n , hence Nrd A (g) = ±µ(g) n .The similitude g is called proper if Nrd A (g) = µ(g) n , and improper if Nrd A (g) = −µ(g) n .The proper similitudes form a subgroup GO + (A, σ) ⊂ GO(A, σ).(As an algebraic group, GO + (A, σ) is the connected component of the identity in GO(A, σ).)Our purpose in this work is to study the multipliers of similitudes of a central simple k-algebra with orthogonal involution (A, σ).We denote by G(A, σ) (resp.G + (A, σ), resp.G − (A, σ)) the group of multipliers of similitudes of (A, σ) (resp.the group of multipliers of proper similitudes, resp.the coset of multipliers of improper similitudes), When A is split (A = End k V for some k-vector space V ), hyperplane reflections are improper similitudes with multiplier 1, hence When A is not split however, we may have G(A, σ) = G + (A, σ).Multipliers of similitudes were investigated in relation with the discriminant disc σ by Merkurjev-Tignol [14].Our goal is to obtain similar results relating multipliers of similitudes to the next invariant of σ, which is the Clifford algebra C(A, σ) (see [7, §8]).As an application, we obtain a complete description of G(A, σ) when deg A ≤ 6 or when the virtual cohomological dimension of k is at most 2.
To give a more precise description of our results, we introduce some more notation.Throughout the paper, k denotes a field of characteristic different from 2. For any integers n, d ≥ 1, let µ 2 n be the group of 2 n -th roots of unity in a separable closure of k and let ) be the d-th cohomology group of the absolute Galois group with coefficients in µ ), so H 1 k and H 2 k may be identified with the 2-primary part of the character group of the absolute Galois group and with the 2-primary part of the Brauer group of k, respectively, In particular, the isomorphism k × /k ×2 ≃ H 1 (k, Z/2Z) derived from the Kummer sequence (see for instance [7, (30.1)]) yields a canonical embedding The Brauer class (or the corresponding element in If K/k is a finite separable field extension, we denote by the norm (or corestriction) map.We extend the notation above to the case where Our results use the product induced as follows by the cup-product: ) and consider the cohomology class (x In particular, if d = 1 and ξ is the square class of y ∈ k × under the embedding (1), then x • ξ is the Brauer class of the quaternion algebra (x, y) k .
Throughout the paper, we denote by A a central simple k-algebra of even degree 2n, and by σ an orthogonal involution of A. Recall from [7, (7.2)] that disc σ ∈ k × /k ×2 ⊂ H 1 k is the square class of (−1) n Nrd A (a) where a ∈ A × is an arbitrary skew-symmetric element.Let Z be the center of the Clifford algebra C(A, σ); thus, Z is a quadratic étale k-algebra, Z = k[ √ disc σ], see [7, (8.10)].The following relation between similitudes and the discriminant is proved in [14, Theorem A] (see also [7, (13.38)]):Theorem 1.Let (A, σ) be a central simple k-algebra with orthogonal involution of even degree.For λ ∈ G(A, σ), Our main results are Theorems 2, 3, 4, and 5 below.

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Theorem 2. Suppose A is split by Z.There exists an element γ(σ) Remark 1.In the conditions of the theorem, the element The last term vanishes in ( The proof of Theorem 2 is given in Section 1.It shows that in the split case, where A = End k V and σ is adjoint to some quadratic form q on V , we may take for γ(σ) the Brauer class of the full Clifford algebra C(V, q).Note that the statement of Theorem 2 does not discriminate between multipliers of proper and improper similitudes, but Theorem 1 may be used to distinguish between them.Slight variations of the arguments in the proof of Theorem 2 also yield the following result on multipliers of proper similitudes: and The proof is given in Section 1.Note however that the theorem holds without the hypothesis that ind A ≤ 4, as follows from Corollaries 1.20 and 1.21 in [12].Using the Rost invariant of Spin groups, these corollaries actually yield an explicit element z as in Theorem 3 from any proper similitude with multiplier λ.
Remark 2. The element and not on the specific choice of z ∈ Z. Indeed, if z, z ′ ∈ Z × are such that N Z/k (z) = N Z/k (z ′ ), then Hilbert's Theorem 90 yields an element u ∈ Z × such that, denoting by ι the nontrivial automorphism of Z/k, Since N Z/k • ι = N Z/k and since the properties of the Clifford algebra (see [7, (9.12) By the projection formula, the right side is equal to Remark 3. Theorems 2 and 3 coincide when they both apply, i.e., if A is split by Z (hence ind A = 1 or 2), and λ ∈ G + (A, σ).Indeed, if λ = N Z/k (z) and γ(σ) Z = [C(A, σ)] then the projection formula yields Remarkably, the conditions in Theorems 1 and 2 turn out to be sufficient for λ to be the multiplier of a similitude when deg A ≤ 6 or when the virtual cohomological 2-dimension3 of k is at most 2.
• If A is not split by Z , then every similitude is proper, Moreover, for λ ∈ k × , we have λ ∈ G(A, σ) if and only if there exists z ∈ Z × such that λ = N Z/k (z) and • If A is split by Z, let γ(σ) ∈ H 2 k be as in Theorem 2. For λ ∈ k × , we have λ ∈ G(A, σ) if and only if The proof is given in Section 2.
Note that if deg A = 2, then A is necessarily split by Z and we may choose γ(σ) = 0, hence Theorem 4 simplifies to For the next statement, recall that the virtual cohomological 2-dimension of . If v is an ordering of k, we let k v be a real closure of k for v and denote simply by (A, σ) v the algebra with involution (A The proof is given in Section 3.
1 Proofs of Theorems 2 and 3 Theorems 2 and 3 are proved by reduction to the split case, which we consider first.We thus assume A = End k V for some k-vector space V of dimension 2n, and σ is adjoint to a quadratic form q on V .Then disc σ = disc q and C(A, σ) is the even Clifford algebra C(A, σ) = C 0 (V, q).We denote by C(V, q) the full Clifford algebra of q, which is a central simple k-algebra, and by I m k the m-th power of the fundamental ideal Ik of the Witt ring W k.
Proof of Theorem 2. If A is split, then using the same notation as in Lemma 1.1 we may take γ(σ) = [C(V, q)], and Theorem 2 readily follows from Lemma 1.1.
For the rest of the proof, we may thus assume A is not split, hence disc σ = 0 since Z is assumed to split A. Let G = {Id, ι} be the Galois group of Z/k.The properties of the Clifford algebra (see for instance [7, (9.12)]) yield Therefore, [C(A, σ)] lies in the subgroup (Br Z) G of Br Z fixed under the action of G.The "Teichmüller cocycle" theory [6] (or the spectral sequence of group extensions, see [19,Remarque,p. 126]) yields an exact sequence hence 4γ(σ) = 0. Therefore, γ(σ) ∈ Br 2 (k) = H 2 k.Note that ind A = 2, since A is split by the quadratic extension Z/k, hence A is Brauer-equivalent to a quaternion algebra Q.Let X be the conic associated with Q; the function field k(X) splits A. Since Theorem 2 holds in the split case, we have By a theorem of (Arason-) Peyre [16,Proposition 4.4], the kernel on the right side is the subgroup Proof of Theorem 3. Suppose first A is split, and use the same notation as in Lemma 1.1.If λ ∈ G(A, σ), then λ • q ≃ q and Lemma 1.1 yields the projection formula yields If A is not split, we extend scalars to the function field k(X) of the Severi-Brauer variety of A. For λ ∈ G + (A, σ), there still exists z ∈ Z × such that λ = N Z/k (z), by Theorem 1.Since Theorem 3 holds in the split case, we have and Peyre's theorem concludes the proof.(Note that applying Peyre's theorem requires the hypothesis that ind A ≤ 4.)

Algebras of low degree
We prove Theorem 4 by considering separately the cases ind A = 1, 2, and 4.

Case 1: A is split
Let A = End k V , dim V ≤ 6, and let σ be adjoint to a quadratic form q on V .Since C(A, σ) = C 0 (V, q), we may choose γ(σ) = [C(V, q)].The equations , and the proof is complete.

Case 2: ind
Let Q be a quaternion (division) algebra Brauer-equivalent to A. We represent A as A = End Q U for some 3-dimensional (right) Q-vector space.The involution σ is then adjoint to a skew-hermitian form h on U (with respect to the conjugation involution on Q), which defines an element in the Witt group Let X be the conic associated with Q.The function field k(X) splits Q, hence Morita equivalence yields an isomorphism Moreover, Dejaiffe [4] and Parimala-Sridharan-Suresh [15] have shown that the scalar extension map is injective.Let (V, q) be a quadratic space over k(X) representing the image of (U, h) under (8).We may assume dim V = deg A ≤ 6 and σ is adjoint to q after scalar extension to k(X).An element λ ∈ k × lies in G(V, q) if and only if λ • q = 0; by the injectivity of ( 8), this condition is also equivalent to Suppose first A is not split by Z. Theorem 1 then shows that every similitude of (A, σ) is proper, and it only remains to show that if λ = N Z/k (z) for some z ∈ Z × such that then λ ∈ G(A, σ).Extending scalars to k(X), we derive from the last equation by the projection formula Therefore, by Lemma 1.1, λ • q ≡ q mod I 4 k(X), i.e., λ • q ∈ I 4 k(X).

Case 3: ind A = 4
Since deg A ≤ 6, this case arises only if deg A = 4, i.e., A is a division algebra.This division algebra cannot be split by the quadratic k-algebra Z, hence all the similitudes are proper, by Theorem 1. Theorem 3 shows that if λ ∈ G(A, σ), then there exists z ∈ Z × such that λ = N Z/k (z) and , and it only remains to prove the converse.

It follows that
hence the form on the left side is isotropic.Therefore, the form u −1 z • z 1 , z 2 represents an element v ∈ k × .Then v −1 u −1 z is represented by z 1 , z 2 , which is the reduced norm form of C(A, σ), hence z ∈ k × Nrd(C(A, σ) × ), and By [7, (15.11)], the group on the right is G + (A, σ).We have thus proved N Z/k (z) ∈ G(A, σ), and the proof is complete when Z is a field.Suppose finally disc σ = 0, i.e., [7, (15.13)] shows We also have z = (z ′ , z ′′ ) for some z ′ , z ′′ ∈ k × , and (10) becomes
To finish this section, we compare the descriptions of G + (A, σ) for deg A = 4 or 6 in [7] with those which follow from Theorem 4 (and Remark 3).
Corollary 2.2.Suppose deg A = 6.If disc σ = 0, let ι be the nontrivial automorphism of the field extension Z/k and let σ be the canonical (unitary) involution of C(A, σ).Let also If disc σ = 0, then C(A, σ) ≃ C × C op for some central simple k-algebra C of degree 4, and

Fields of low virtual cohomological dimension
Our goal in this section is to prove Theorem 5. Together with Theorem 2, the following lemma completes the proof of the "only if" part: Proof.If (A, σ) v is not hyperbolic, then A v is split, by [18, Chapter 10, Theorem 3.7].We may thus represent A v = End kv V for some k v -vector space V , and σ ⊗ Id kv is adjoint to a non-hyperbolic quadratic form q. If λ ∈ G(A, σ), then λ ∈ G(V, q), hence λ • q ≃ q.
Comparing the signatures of each side, we obtain λ > 0.
Proof of Theorem 5.As observed above, the "only if" part follows from Theorem 2 and Lemma 3.1.The proof of the "if" part uses the same arguments as the proof of Theorem 2 in the case where ind A = 2.We first consider the split case.If A = End k V and σ is adjoint to a quadratic form q on V , then we may choose γ(σ) = C(V, q), and the conditions imply, by Lemma 1.1, that λ • q ∈ I 4 k.Moreover, for every ordering v on k, the signature sgn v ( λ • q) vanishes, since λ > 0 at every v such that sgn v (q) = 0. Therefore, by Pfister's local-global principle [8, Chapter 8, Theorem 4.1], λ • q is torsion.Since the hypothesis on k implies, by Lemma 3.2, that I 4 t k = 0, we have λ • q = 0, hence λ ∈ G(V, q) = G(A, σ).Note that Lemma 3.2 yields I 4 t k = 0 under the weaker hypothesis vcd 2 k ≤ 3. Therefore, the split case of Theorem 5 holds when vcd 2 k ≤ 3. Now, suppose A is not split.Since A is split by Z, it is Brauer-equivalent to a quaternion algebra Q.Let k(X) be the function field of the conic X associated with Q.This field splits A, hence there is a quadratic space (V, q) over k(X) such that A ⊗ k(X) may be identified with End k(X) V and σ ⊗ Id k(X) with the adjoint involution with respect to q.As in Section 2 (see Equation ( 9)), we have Therefore, it suffices to show that the conditions on λ imply λ ∈ G(V, q).If v is an ordering of k such that (A, σ) v is hyperbolic, then q w is hyperbolic for any ordering w of k(X) extending v, since hyperbolic involutions remain hyperbolic over scalar extensions.Therefore, λ > 0 at every ordering w of k(X) such that q w is not hyperbolic.Moreover, the conditions Since X is a conic, Proposition 11, p. 93 of [20] implies As Theorem 5 holds in the split case over fields of virtual cohomological 2dimension at most 3, it follows that λ ∈ G(V, q).
Remark.The same arguments show that if vcd 2 k ≤ 2 and ind A = 2, then G + (A, σ) consists of the elements N Z/k (z) where z ∈ Z × is such that

Examples
In this section, we give an explicit description of the element γ(σ) of Theorem 2 in some special cases.Throughout this section, we assume the algebra A is not split, and is split by Z (hence Z is a field and disc σ = 0).Our first result is easy: Proposition 4.1.If A is split by Z and σ becomes hyperbolic after scalar extension to Z, then we may choose γ(σ) = 0.
Proof.Let ι be the nontrivial automorphism of Z/k.Since Z is the center of On the other hand, C(A, σ) Proof.This readily follows from Proposition 4.1 and Theorem 2 or 5.
To give further examples where γ(σ) can be computed, we fix a particular representation of A as follows.Since A is assumed to be split by Z, it is Brauer-equivalent to a quaternion k-algebra Q containing Z.We choose a quaternion basis 1, i, j, ij of Q such that Z = k(i).Let A = End Q U for some right Q-vector space U , and let σ be the adjoint involution of a skew-hermitian form h on U with respect to the conjugation involution on Q.For x, y ∈ U , we decompose h(x, y) = f (x, y) + jg(x, y) with f (x, y), g(x, y) ∈ Z.
It is easily verified that f (resp.g) is a skew-hermitian (resp.symmetric bilinear) form on U viewed as a Z-vector space.(See [18, Chapter 10, Lemma 3.1].)We have Moreover, for x, y ∈ U and ϕ ∈ End Q U , the equation Proof.Since σ ⊗ Id Z is the adjoint involution of g, Now, disc σ is a square in Z, hence C 0 (U, g) decomposes into a direct product where C ′ , C ′′ are central simple Z-algebras Brauer-equivalent to C(U, g).The proposition follows from (11), (12), and (13).
To give an explicit description of g, consider an h-orthogonal basis (e 1 , . . ., e n ) of U .In the corresponding diagonalization of h, so we may assume i 2 = a 1 . . .a n .Write Each e ℓ Q is a 2-dimensional Z-vector space, and we have a g-orthogonal decomposition If v ℓ = 0, then g(e ℓ , e ℓ ) = 0, hence e ℓ Q is hyperbolic.If v ℓ = 0, then (e ℓ , e ℓ u ℓ ) is a g-orthogonal basis of e ℓ Q, which yields the following diagonalization of the restriction of g: v ℓ , −a ℓ v ℓ . Therefore, where We now consider in more detail the cases n = 2 and n = 3.

Algebras of degree 4
Suppose deg A = 4, i.e., n = 2, and use the same notation as above.If v 1 = 0, then squaring each side of ( 14) yields a 1 = µ 2 1 a 1 a 2 , hence a 2 ∈ k ×2 , a contradiction since Q is assumed to be a division algebra.The case v 2 = 0 leads to the same contradiction.Therefore, we necessarily have v 1 = 0 and v 2 = 0.By ( 15) and ( 16), hence by [8, p. 121 Since the division algebra Q contains the pure quaternions u 1 , u 2 and i with u 2 1 = a 1 , u 2 2 = a 2 and i 2 = a 1 a 2 , we have a 1 , a 2 , a 1 a 2 / ∈ k ×2 and we may consider the field extension We identify Z with a subfield of L by choosing in L a square root of a 1 a 2 , and denote by ρ 1 , ρ 2 the automorphisms of L/k defined by Thus, Z ⊂ L is the subfield of ρ 1 • ρ 2 -invariant elements.Let j 2 = b.Then ( 14) yields ).

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Since L = Z( √ a 1 ) = Z( √ a 2 ), it follows that 1 − µ 2 1 a 2 and 1 − µ 2 2 a 1 are norms from L/Z.Therefore, the preceding equation yields Define an algebra E over k by where the multiplication is defined by Since b 1 , b 2 and ℓ satisfy ( 18) and ( 19), the algebra E is a crossed product, see [1].It is thus a central simple k-algebra of degree 4. Proof.The centralizer Since conjugation by r 1 r 2 maps √ a 1 ∈ L to its opposite, it follows that Z , the proposition follows from (17).
To prove From the definition of E, it follows that r 1 ∈ E − .By [10, p. 80], Let L 1 ⊂ L be the subfield fixed under ρ 1 .We have r On the other hand, the centralizer of L 1 is Since , hence (21) yields From ( 20), (22) and Theorems 1, 2 it follows that Nrd E (r 1 ) ∈ G − (A, σ).Now, suppose x ∈ E − .Then r 1 x ∈ E + , hence Nrd E (r 1 x) ∈ G + (A, σ) by the first part of the corollary.Since We have thus proved we have λ Nrd E (r 1 ) ∈ G + (A, σ), hence by the first part of the corollary,

Algebras of degree 6
Suppose deg A = 6, i.e., n = 3, and use the same notation as in the beginning of this section.If σ (i.e., h) is isotropic, then h is Witt-equivalent to a rank 1 skew-hermitian form, say u .Hence i 2 = disc σ = u 2 ∈ k × .Hence we may assume that h is Witt-equivalent to the rank 1 skew-hermitian form µi for some µ ∈ k × .This implies that the form g is hyperbolic and C(U, g) is split.Hence we may choose γ(σ) = 0.By Theorem 4, we then have λ ∈ G(A, σ) if and only if λ. disc σ = 0 in (H 2 k)/A.If σ becomes isotropic over Z, the form g is isotropic, hence we may choose a diagonalization of h h ≃ u 1 , u 2 , u 3 such that g(u 3 , u 3 ) = 0, i.e., in the notation of ( 14), u 3 = µ 3 i.Raising each side to the square, we obtain hence a 1 ≡ a 2 mod k ×2 .It follows that u 2 is conjugate to a scalar multiple of u 1 , i.e., there exists x ∈ Q × and θ ∈ k × such that Since u 1 ≃ xu 1 x , we may let ν = −θ Nrd(x) −1 ∈ k × to obtain h ≃ u 1 , −νu 1 , µ 3 i .